All exercise S.Qs of Chapter#3: Time Dependent Perturbation Theory of Quantum Mechanics-II. We have arranged all S.Qs for BS/MSc Physics students.
Q.3.1 why transition probability at $\omega_fi=\omega$ act as absorption for harmonic perturbation?
Answer:
$\omega_{fi} -\ \omega=0$
$Ef\ -Ei =\hbar\omega$
The perturbation can induce transitions from Ei only to a level Ef whose energy is higher than EI by just . Such a transition may be described as absorption energy $\hbar\omega$ by the system.
Q.3.2 why energy is conserved in $\delta(Ef-\ Ei)$ in constant perturbation?
Answer: The delta term guarantees the conservation of energy: in the limit $t\rightarrow\infty$, the transition rate is non-vanishing only between states of equal energy. Hence, a constant (time independent) perturbation neither removes energy from the system nor supplies energy to it. It simply causes energy-conserving transitions.
Q.3.3 what is the time period of transition probability of oscillating sinusoidal function?
Answer:
$$P_{fi}\left(t\right)=\frac{v^2_{fi}}{\hbar} \left[\frac{{4sin}^{2\ }\ (\omega_{fi\ }-\omega)}{(\omega_{fi\ }+\omega)\ ^2}\ \frac{t}{2}\right]$$
Or
$$P_{fi}(t)=\left | -\frac{i}{\hbar}\int_{0}^{t}< \psi _{f}|\hat{V}(t’)|\psi _i> e^{i\omega fit’}dt’ \right |^2$$
This is time period.
Q.3.4 why absorption and emission process cannot take place simultaneously?
Answer: The absorption and emission are inverse process.
Absorption process:
When atoms absorbs energy of photon to promote electron to higher energy orbital;
Emission process:
When atoms emits energy as photon as electron falls from higher energy orbital to lower;
Q.3.5 wrote down necessary conditions for the existence of Pif for a harmonic perturbation.
Answer: The transition rate $(\frac{P_{if} }{t}=$
The first condition $E_f=E_{i}- \hbar \omega$ is called emission.
The second condition $E_f=\ E_{i\ } + \hbar \omega$ is called absorption.
Q.3.6 why transition probability has an interference pattern?
Answer: As a function of $\omega_{fi}$ , the transition probability has an interference pattern. It is appreciable only near $\omega_{fi} = 0$ and decays moves away from zero.
Q.3.7 Define MASER and LASER.
Answer: MASER: (Microwave amplification by stimulated emission of radiation).
LASER: (Light amplification by stimulated emission of radiation).
